objects with non-alternating simple symmetry group

Question

The usual soccer ball has as its group of symmetries (counting rotations only) the alternating group $A_5$. Is there an example of a well known object whose group of symmetries is a finite simple group of Lie type over a finite field? Obviously I want to avoid the case when the group of Lie type is isomorphic to an alternating group.

Answer 1

It is possible to inscribe a regular tetrahedron in a cube, but not an equilateral triangle in a square. In general, we can ask in which dimensions $d$ it is possible to find a $d$-simplex within the vertices of a $d$-cube.

It turns out this is possible only when $d\equiv 3\pmod{4}$, and it is conjectured that it can be done for all such $d$, though we only know this for $d<667$. (See Hadamard matrices for more information here.)

When $d=7$, there is a unique way up to rotation to inscribe this $7$-simplex; the symmetries of the cube fixing this simplex form a group of order $1344$, and the subgroup fixing a vertex of the simplex forms the simple group $PSL_2(7)$ of order $168$, also the symmetry group of the Fano plane.

When $d=11$, there is again a unique way to position this simplex up to rotation; its symmetries form the sporadic simple group $M_{11}$ (and in fact induce its transitive action on $12$ points via the vertices of the $11$-simplex), and the pointwise stabilizer of this action forms the simple group $PSL_2(11)$ of order $660$, also the symmetry group of the Paley biplane. (The stabilizer of an additional point gives us the group $PSL_2(5)\cong A_5$, the symmetries of an icosahedron.)

In fact, we can extend the above embedding of an $11$-simplex into an $11$-cube into an embedding of a $12$-orthoplex into a $12$-cube, where the symmetry group (after we quotient out by the $C_2$ center) is the sporadic simple group $M_{12}$.

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Beyond the Fano plane, some larger projective planes contain various simple groups as subgroups of their symmetry groups - I haven't looked into this much, but I would guess the subgroups found in this manner are the stabilizers of relatively natural structures within these projective planes.